I wanted to add this to the original post but can't reply there so I am starting a new post here.
Yesterday evening (Sunday around 11 P.M. GMT +1) it happened again, just as described above. I noticed some more things this time:
1. As before, I tried to place a GM call but didn't get a response so I assume you can't contact GMs properly while suffering from the bug. When I tried to contact a GM the last time, I didn't get a response on the same evening, but when I logged in the next day, I had an empty (!) Help Desk message. Another day later there was another message, telling me that when they tried to reach me, I wasn't online. Something seems to be really messed up with GM calls there.
2. Before opening the chest in the end, I logged out and switched to a different character. That one was in the Mog House and seemed to be affected by the same bug since no emotes were displayed (I didn't try anything else but I assume it would have been the same with battle messages etc.).
3. I noticed that the bug always occurs when I am the party leader who started the Skirmish operation.
4. I made a file check, just to be sure there is nothing wrong with the data installed on my Xbox, but neither POL nor FFXI produced any errors there.